Algorithms for Nonnegative $C^2(\mathbb{R}^2)$ Interpolation

TL;DR

This paper presents efficient algorithms for nonnegative $C^2$ interpolation in the plane, enabling the extension of discrete data to smooth, nonnegative functions with near-optimal norm control.

Contribution

It introduces novel algorithms for constructing nonnegative $C^2$ extensions and approximating trace norms, advancing computational methods in shape-preserving interpolation.

Findings

Algorithms run efficiently for finite sets in the plane.

Extensions maintain nonnegativity and smoothness with near-optimal norm.

Provides practical tools for shape-preserving data interpolation.

Abstract

Let $ E \subset \mathbb{R}^2 $ be a finite set, and let $ f : E \to [0,\infty) $. In this paper, we address the algorithmic aspects of nonnegative $C^2$ interpolation in the plane. Specifically, we provide an efficient algorithm to compute a nonnegative $C^2(\mathbb{R}^2)$ extension of $ f $ with norm within a universal constant factor of the least possible. We also provide an efficient algorithm to approximate the trace norm.